PARADISE LOST? point P traverses CD, while Q simultaneously traverses AE, and each point of CD has one, and only one, 'paired** point of AB. An even more unexpected result can be proved. Any line- segment 5 no matter how small, contains as many points as an infinite straight line. Further, the segment contains as many points as there are in an entire plane, or in the whole of three- dimensional space, or in the whole of space of n dimensions (where n is any integer greater than zero) or, finally, in a space of a denumerably infinite number of dimensions. In all this we have not yet attempted to define a class or a set. Possibly (as Russell held in 1912) it is not necessary to do so in order to have a clear conception of Cantor's theory or for that theory to be consistent with itself - which is enough to demand of any mathematical theory. Nevertheless present disputes seem to require that some clear9 self-consistent definition be given. The following used to be thought satisfactory. A set is characterized by three qualities: it contains all things iio which a certain definite property (say redness, or volume, or taste) belongs; no thing not having this property belongs to the set; each thing in the set is recognizable as the same thing and as different from all other things in the set - briefly, each thing hi the set has a permanently recognizable individuality. The set itself is to be grasped as a whole. This definition may be too drastic for use. Consider, for example, what happens to Cantor's set of all transcendental numbers tinder the third demand* At this point we may glance back over the whole history of mathematics - or as much of it as is revealed by the treatises of the master mathematicians in their purely technical works - and note two modes of expression which recur constantly in nearly all mathematical exposition. The reader perhaps hag been irritated by the repetitious use of phrases such as fiwe can find a whole number greater than 2% or *we can choose a number less than n and greater than n — 2.' The choice of such phrase- ology is not merely stereotyped pedantry. There is a reason for its use, and careful writers mean exactly what they say when they assert that Sre can find, etc*. They mean that they can do what they say. 631